2021 Fall AMC 12B Problems/Problem 11
- The following problem is from both the 2021 Fall AMC 10B #14 and 2021 Fall AMC 12B #11, so both problems redirect to this page.
Contents
Problem
Una rolls standard -sided dice simultaneously and calculates the product of the numbers obtained. What is the probability that the product is divisible by
Solution
We will use complementary counting to find the probability that the product is not divisible by . Then, we can find the probability that we want by subtracting this from 1. We split this into cases.
Case 1: The product is not divisible by .
We need every number to be odd, and since the chance we roll an odd number is our probability is
Case 2: The product is divisible by , but not by .
We need numbers to be odd, and one to be divisible by , but not by . There is a chance that an odd number is rolled, a chance that we roll a number satisfying the second condition (only and work), and ways to choose the order in which the even number appears.
Our probability is
Therefore, the probability the product is not divisible by is .
Our answer is .
~kingofpineapplz (Most of Solution)
~stjwyl (Edits)
Video Solution by Interstigation
https://www.youtube.com/watch?v=G_31gUNwkzQ
Video Solution (Just 4 min!)
~Education, the Study of Everything
Video Solution by WhyMath
~savannahsolver
Video Solution by TheBeautyofMath
https://youtu.be/R7TwXgAGYuw?t=833
~IceMatrix
See Also
2021 Fall AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 10 |
Followed by Problem 12 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.